Now, navigate to the corresponding web site, Site Actions, Site Permissions, and Anonymous Access. Configure what anonymous users can access at the site level

Break the permission inheritance on the specific list in which you want to allow anonymous users to add items.

Navigate to the list/library settings for the list in which you want to allow Anonymous users to add items. Navigate to List Settings, Permissions for this list, click on Stop Inheriting Permissions and click OK. Click on Anonymous Access and select Add Items.

Enable “AddItem” operation for the client object model for the Web Application. You can run the following code as a console application (equivalent PowerShell script also should work).